Instructor: Tadahiro Oh (University of Edinburgh & Beijing Institute of Technology)
Organizers: Xicheng Zhang, Rongchan Zhu, Xiangchan Zhu, and Guopeng Li
Time: 19:00 - 21:00 in Beijing (11:00 - 13:00 in the UK) on Tuesdays,
November, 2024 -
Course description:Dispersive PDEs such as the nonlinear Schrodinger equations (NLS) and the nonlinear wave equations (NLW) appear ubiquitously in applications, describing wave propagations in various physical contexts. Over the last forty years, multilinear harmonic analysis has played an essential role in the development of theoretical understanding of nonlinear dispersive PDEs, in particular in the low regularity setting.Furthermore, over the last fifteen years, there has been remarkable progress in the study lying at the intersection of nonlinear dispersive PDEs and probability theory.In this course, we go over basic well-posedness theory of dispersive PDEs from both the deterministic and probabilistic viewpoints.
Topics include:
（1）. Basic well-posedness theory of NLS on Rd via the Strichartz estimate: oscillatory integral approach.
（2）. Fourier restriction norm method: local well-posedness of NLS on Td and KdV on T.
（3）. Construction of Gibbs measures and invariant Gibbs dynamics.
（4）. Probabilistic well-posedness theory for random dispersive PDEs with random initial data and/or stochastic forcing.
Further topics may include:
● paracontrolled approach to NLW (Gubinelli, Koch, and Oh, J. Eur. Math. Soc. (2024))
● random averaging operators (Deng, Nahmod, and Yue, Ann. of Math. (2024)) among others (which may be discussed in the subsequent years).
Supplementary reading:Here, I list some supplementary reading materials:
● My course notes on "Nonlinear Schrodinger equations "(2018).
● Notes on "Nonlinear Schrodinger equations" (2020)
Also, HW and Takehome final with solutions (which are more like lecture notes): HW1，HW2，HW3，Takehome final
● My course notes on "Probabilistic Perspectives in Nonlinear Dispersive PDEs" (2017).
● My course notes on "Singular stochastic dispersive PDEs" (2021).
Please make sure to read the notes hand-written by me (and not the notes typed by students, which may have typos and errors). You may find other notes on my "website" under “Teaching” and under“Notes”(below my papers) useful.
